Harmonic vibration of a damped system

by Yi Zhang

Unlike in previous post that phase change could only be or , in the damped system, as the ratio between dynamics response and static response is expressed in the same way as the undamped case
the phase lag varies in . The following Mathematica outputs the displacement response factor, phase lag and normalized time history as the function of damping ratio and frequency ratio .

Long period excitation, i.e., , gives a pseudo-static response. In this case, the system “waits” until it “feels” the excitation completely. is greater but very close to , and the displacement is essentially in phase with excitation force, in other words, dynamic effect is near to none.

Long period/low frequency loading response

Short period excitation gives very small , though the phase lag is . Here the system barely reacts when the load is reversed, thus leads to small displacement.

Short period/high frequency loading response

Resonant period, i.e., , leads to . Now is very sensitive to damping change, namely, the response is controlled by the damping: a small change of damping ratio leads to great reaction of the structure. When , we have .

Resonant frequency: small damping

Resonant frequency: medium damping

Resonant frequency: large damping

The last case is the what’s essentially behind the viscous damping devices applied to buildings like this.